Super-relational fixed point property and products in ordered sets (Q925254)

Let \(P\) be an ordered set and \(E(P)\) be the set of all endomorphisms (isotone mappings) of \(P\) into \(P\). If \(f\in E(P)\), denote by \(\text{Fix}(f)\) the set of all fixed points of \(f\), i.e., \(\text{Fix}(f):=\{p\in P:f(p)=p\}\). An ordered set \(P\) has the fixed-point property if \(\text{Fix}(f)\not=\emptyset\) for every \(f\in E(P)\). In the paper, several stronger variants of the fixed-point property are summarized and the connections among them are described. Furthermore, a new kind of such a stronger property for ordered sets, called the super-relational fixed-point property, is introduced and it is compared with strengthened and product fixed-point properties.